prove sec(x*0)=1

Solution

prove

sec (x \cdot 0) = 1

True

Solution steps

sec (x \cdot 0) = 1

Manipulating left side

sec (x 0)

Simplify

sec (0 x) : 1

sec (0 \cdot x)

Apply rule

0 \cdot a = 0

= sec (0)

Use the following trivial identity:

sec (0) = 1

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( 2 x )}{sin ^{2} ( x )} = csc^{2} (x) - 2

\frac{2 cos ( x )}{csc ( x ) - 2 sin ( x )} = tan (2 x)

\frac{1 + csc ( a )}{cot ( a ) + cos ( a )} = sec (a)

sec (θ) - sec (θ) sin^{2} (θ) = cos (θ)

- 2 = - 2 cos (0) + 15 sin (3) (0)